Derivative Rules Cheat Sheet

A comprehensive set of vocabulary flashcards covering basic, product, quotient, chain, trigonometric, inverse trigonometric, exponential, and logarithmic derivative rules.

Constant Rule

The derivative of a constant is zero, expressed as $\frac{d}{dx}(c) = 0$.

Power Rule

A rule where you bring down the exponent and reduce it by 1, expressed as $\frac{d}{dx}(x^n) = nx^{n-1}$.

Constant Multiple

A rule where constants factor out of the differentiation, expressed as $\frac{d}{dx}(cf(x)) = c \cdot f'(x)$.

Sum Rule

The derivative of a sum is equal to the sum of the derivatives, expressed as $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$.

Difference Rule

The derivative of a difference is equal to the difference of the derivatives, expressed as $\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)$.

Product Rule

Defined as the first function times the derivative of the second plus the second function times the derivative of the first, expressed as $\frac{d}{dx}(f(x) \cdot g(x)) = f'(x)g(x) + f(x)g'(x)$.

Quotient Rule

Defined as low d-high minus high d-low over low squared, expressed as $\frac{d}{dx}(\frac{f(x)}{g(x)}) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$.

Chain Rule

The derivative of the outer function times the derivative of the inner function, expressed as $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$.

General Power

The application of the power rule combined with the chain rule, expressed as $\frac{d}{dx}(u^n) = nu^{n-1} \cdot \frac{du}{dx}$.

Sine

The trigonometric derivative expressed as $\frac{d}{dx}(\sin(x)) = \cos(x)$.

Cosine

The trigonometric derivative expressed as $\frac{d}{dx}(\cos(x)) = -\sin(x)$.

Tangent

The trigonometric derivative expressed as $\frac{d}{dx}(\tan(x)) = \sec^2(x)$.

Cotangent

The trigonometric derivative expressed as $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$.

Secant

The trigonometric derivative expressed as $\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$.

Cosecant

The trigonometric derivative expressed as $\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$.

Arcsine

The inverse trigonometric derivative expressed as $\frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1-x^2}}$.

Arccosine

The inverse trigonometric derivative expressed as $\frac{d}{dx}(\arccos(x)) = -\frac{1}{\sqrt{1-x^2}}$.

Arctangent

The inverse trigonometric derivative expressed as $\frac{d}{dx}(\arctan(x)) = \frac{1}{1+x^2}$.

Natural Exponential

The derivative of the exponential function with base $e$, expressed as $\frac{d}{dx}(e^x) = e^x$.

General Exponential

The derivative of an exponential function with base $a$, expressed as $\frac{d}{dx}(a^x) = a^x \ln(a)$.

Natural Log

The derivative of the natural logarithm, expressed as $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$.

General Log

The derivative of a logarithm with base $a$, expressed as $\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$.